Simplify cos (sin −1 x ) and tan(sin −1 x ).

6. To compute the derivatives of the inverse trigonometric functions, we will need to simplify composite expressions such as cos(sin−1x) and tan(sec−1x). This can be done in two ways: by referring to the appropriate right triangle or by using trigonometric identities.

7. Simplifycos(sin−1x) and tan(sin−1x).

8. Derivatives of Arcsine and Arccosine THEOREM 1 Derivatives of Arcsine and Arccosine

9. f (x) = arcsin(x2)

13. THEOREM 2Derivatives of Inverse Trigonometric Functions

14. THEOREM 2Derivatives of Inverse Trigonometric Functions

15. The formulas for the derivatives of the inverse trigonometric functions yield the following integration formulas. Integral Formulas In this list, we omit the integral formulas corresponding to the derivatives of cos−1x,cot−1x, and csc−1x

16. We can use these formulas to express the inverse trigonometric functions as definite integrals. For example, because sin−1 0 = 0, we have:

18. Using Substitution

19. Using Substitution